Ising Model Abstract Art

The Ising Model

From ferromagnets to neural networks: explore how simple local interactions create complex global order.

science Launch Simulation

tune Lab Controls

2.27
0 (Frozen) 5 (Hot)
0.00
-1 (Down) +1 (Up)
Spin Up (+1)
Spin Down (-1)
lightbulb

At the critical temperature (Tc ≈ 2.27), the system undergoes a phase transition. Look for large "fractal-like" clusters of spins forming and dissolving.

monitoring Real-time Data

Net Magnetization (M) 0.00
-1 (All Down) 0 +1 (All Up)

Magnetization History

Energy/Spin
-1.45
Acceptance
12%

The Mathematics of Magnetism

The Ising model simplifies a magnet into a grid of atomic "spins" that can point either up (+1) or down (-1). Atoms want to align with their neighbors (ferromagnetism) and with any external magnetic field.

// Hamiltonian (Total Energy)

H(σ) = -J ∑ σiσj - B ∑ σi

σ : Spin state (+1 or -1)
J : Interaction strength
B : External Field
: Sum over neighbors

The Phase Transition

At high temperatures, thermal noise flips spins randomly, destroying order (Paramagnetic phase). At low temperatures, neighbor interactions win, creating large domains of aligned spins (Ferromagnetic phase). The sharp change occurs at the Critical Temperature (Tc).

Fact: For a 2D square lattice, Onsager proved Tc is exactly 2 / ln(1 + √2) ≈ 2.269.
Iron Filings Magnetism

Iron filings revealing magnetic field lines.

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